exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 13.0s

Alternatives: 9

Speedup: N/A×

• could not determine a ground truth

  1. w = 2.00000000000000007e154
  2. l = -2

Specification

Math

\[\begin{array}{l} \\ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (exp (- w)) (pow l (exp w))))

C

double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp(-w) * (l ** exp(w))
end function

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

Julia

function code(w, l)
	return Float64(exp(Float64(-w)) * (l ^ exp(w)))
end

MATLAB

function tmp = code(w, l)
	tmp = exp(-w) * (l ^ exp(w));
end

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (exp (- w)) (pow l (exp w))))

C

double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp(-w) * (l ** exp(w))
end function

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

Julia

function code(w, l)
	return Float64(exp(Float64(-w)) * (l ^ exp(w)))
end

MATLAB

function tmp = code(w, l)
	tmp = exp(-w) * (l ^ exp(w));
end

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ (pow l (exp w)) (exp w)))

C

double code(double w, double l) {
	return pow(l, exp(w)) / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = (l ** exp(w)) / exp(w)
end function

Java

public static double code(double w, double l) {
	return Math.pow(l, Math.exp(w)) / Math.exp(w);
}

Python

def code(w, l):
	return math.pow(l, math.exp(w)) / math.exp(w)

Julia

function code(w, l)
	return Float64((l ^ exp(w)) / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = (l ^ exp(w)) / exp(w);
end

Wolfram

code[w_, l_] := N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.3%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. remove-double-neg 99.3%

      \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   3. associate-*l/ 99.3%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

   4. *-lft-identity 99.3%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

   5. remove-double-neg 99.3%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 97.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{e^{w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ l (exp w)))

C

double code(double w, double l) {
	return l / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l / exp(w)
end function

Java

public static double code(double w, double l) {
	return l / Math.exp(w);
}

Python

def code(w, l):
	return l / math.exp(w)

Julia

function code(w, l)
	return Float64(l / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = l / exp(w);
end

Wolfram

code[w_, l_] := N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.3%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. remove-double-neg 99.3%

      \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   3. associate-*l/ 99.3%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

   4. *-lft-identity 99.3%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

   5. remove-double-neg 99.3%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 43.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

   2. sqrt-unprod 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

   3. sqr-neg 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

   4. sqrt-unprod 41.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

   5. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

   6. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

   7. sqrt-unprod 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

   8. add-sqr-sqrt 41.3%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   9. sqrt-unprod 70.5%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   10. sqr-neg 70.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   11. sqrt-unprod 29.2%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   12. add-sqr-sqrt 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

   13. pow1 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

   14. exp-neg 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

   15. inv-pow 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

   16. pow-prod-up 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

   17. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

   18. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

   19. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]

7. Taylor expanded in l around 0 98.3%

   \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]
8. Add Preprocessing

Alternative 3: 76.8% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 - w \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (* l (+ 1.0 (* w (+ -1.0 (* w (- 0.5 (* w 0.16666666666666666))))))))

C

double code(double w, double l) {
	return l * (1.0 + (w * (-1.0 + (w * (0.5 - (w * 0.16666666666666666))))));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 + (w * ((-1.0d0) + (w * (0.5d0 - (w * 0.16666666666666666d0))))))
end function

Java

public static double code(double w, double l) {
	return l * (1.0 + (w * (-1.0 + (w * (0.5 - (w * 0.16666666666666666))))));
}

Python

def code(w, l):
	return l * (1.0 + (w * (-1.0 + (w * (0.5 - (w * 0.16666666666666666))))))

Julia

function code(w, l)
	return Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * Float64(0.5 - Float64(w * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 + (w * (-1.0 + (w * (0.5 - (w * 0.16666666666666666))))));
end

Wolfram

code[w_, l_] := N[(l * N[(1.0 + N[(w * N[(-1.0 + N[(w * N[(0.5 - N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.3%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. remove-double-neg 99.3%

      \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   3. associate-*l/ 99.3%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

   4. *-lft-identity 99.3%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

   5. remove-double-neg 99.3%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 43.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

   2. sqrt-unprod 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

   3. sqr-neg 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

   4. sqrt-unprod 41.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

   5. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

   6. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

   7. sqrt-unprod 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

   8. add-sqr-sqrt 41.3%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   9. sqrt-unprod 70.5%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   10. sqr-neg 70.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   11. sqrt-unprod 29.2%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   12. add-sqr-sqrt 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

   13. pow1 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

   14. exp-neg 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

   15. inv-pow 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

   16. pow-prod-up 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

   17. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

   18. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

   19. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]
7. Step-by-step derivation

   1. *-rgt-identity 98.3%

      \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   2. frac-2neg 98.3%

      \[\leadsto \color{blue}{\frac{-\ell}{-e^{w}}} \]

   3. div-inv 98.3%

      \[\leadsto \color{blue}{\left(-\ell\right) \cdot \frac{1}{-e^{w}}} \]

8. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(-\ell\right) \cdot \frac{1}{-e^{w}}} \]

9. Taylor expanded in w around 0 79.7%

   \[\leadsto \left(-\ell\right) \cdot \color{blue}{\left(w \cdot \left(1 + w \cdot \left(0.16666666666666666 \cdot w - 0.5\right)\right) - 1\right)} \]

10. Final simplification 79.7%

    \[\leadsto \ell \cdot \left(1 + w \cdot \left(-1 + w \cdot \left(0.5 - w \cdot 0.16666666666666666\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 76.8% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.8333333333333334\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (+ l (* l (* w (+ (* w (+ 0.5 (* w -0.8333333333333334))) -1.0)))))

C

double code(double w, double l) {
	return l + (l * (w * ((w * (0.5 + (w * -0.8333333333333334))) + -1.0)));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (l * (w * ((w * (0.5d0 + (w * (-0.8333333333333334d0)))) + (-1.0d0))))
end function

Java

public static double code(double w, double l) {
	return l + (l * (w * ((w * (0.5 + (w * -0.8333333333333334))) + -1.0)));
}

Python

def code(w, l):
	return l + (l * (w * ((w * (0.5 + (w * -0.8333333333333334))) + -1.0)))

Julia

function code(w, l)
	return Float64(l + Float64(l * Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.8333333333333334))) + -1.0))))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (l * (w * ((w * (0.5 + (w * -0.8333333333333334))) + -1.0)));
end

Wolfram

code[w_, l_] := N[(l + N[(l * N[(w * N[(N[(w * N[(0.5 + N[(w * -0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.3%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. remove-double-neg 99.3%

      \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   3. associate-*l/ 99.3%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

   4. *-lft-identity 99.3%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

   5. remove-double-neg 99.3%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 43.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

   2. sqrt-unprod 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

   3. sqr-neg 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

   4. sqrt-unprod 41.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

   5. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

   6. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

   7. sqrt-unprod 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

   8. add-sqr-sqrt 41.3%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   9. sqrt-unprod 70.5%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   10. sqr-neg 70.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   11. sqrt-unprod 29.2%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   12. add-sqr-sqrt 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

   13. pow1 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

   14. exp-neg 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

   15. inv-pow 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

   16. pow-prod-up 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

   17. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

   18. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

   19. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]

7. Taylor expanded in w around 0 76.8%

   \[\leadsto \color{blue}{\ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-out 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\ell \cdot \left(-0.5 + 0.16666666666666666\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   2. add-sqr-sqrt 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   3. sqrt-unprod 58.6%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\sqrt{\ell \cdot \ell}} \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   4. sqr-neg 58.6%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \sqrt{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   5. sqrt-unprod 0.0%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\left(\sqrt{-\ell} \cdot \sqrt{-\ell}\right)} \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   6. add-sqr-sqrt 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\left(-\ell\right)} \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   7. cancel-sign-sub-inv 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   8. mul-1-neg 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\color{blue}{\left(-\left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   9. distribute-rgt-out 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(-\color{blue}{\ell \cdot \left(-1 + 0.5\right)}\right) - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   10. metadata-eval 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(-\ell \cdot \color{blue}{-0.5}\right) - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   11. distribute-rgt-neg-in 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\color{blue}{\ell \cdot \left(--0.5\right)} - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   12. metadata-eval 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot \color{blue}{0.5} - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   13. metadata-eval 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot 0.5 - \ell \cdot \color{blue}{-0.3333333333333333}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

9. Applied egg-rr 76.8%

   \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell \cdot 0.5 - \ell \cdot -0.3333333333333333\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]
10. Step-by-step derivation

    1. distribute-lft-out-- 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell \cdot \left(0.5 - -0.3333333333333333\right)\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

    2. metadata-eval 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot \color{blue}{0.8333333333333334}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

11. Simplified 76.8%

    \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell \cdot 0.8333333333333334\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

12. Taylor expanded in l around 0 79.7%

    \[\leadsto \ell + \color{blue}{\ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + -0.8333333333333334 \cdot w\right) - 1\right)\right)} \]

13. Final simplification 79.7%

    \[\leadsto \ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.8333333333333334\right) + -1\right)\right) \]
14. Add Preprocessing

Alternative 5: 73.8% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 + w \cdot \left(-1 - w \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (+ 1.0 (* w (- -1.0 (* w -0.5))))))

C

double code(double w, double l) {
	return l * (1.0 + (w * (-1.0 - (w * -0.5))));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 + (w * ((-1.0d0) - (w * (-0.5d0)))))
end function

Java

public static double code(double w, double l) {
	return l * (1.0 + (w * (-1.0 - (w * -0.5))));
}

Python

def code(w, l):
	return l * (1.0 + (w * (-1.0 - (w * -0.5))))

Julia

function code(w, l)
	return Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 - Float64(w * -0.5)))))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 + (w * (-1.0 - (w * -0.5))));
end

Wolfram

code[w_, l_] := N[(l * N[(1.0 + N[(w * N[(-1.0 - N[(w * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.3%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. remove-double-neg 99.3%

      \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   3. associate-*l/ 99.3%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

   4. *-lft-identity 99.3%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

   5. remove-double-neg 99.3%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 43.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

   2. sqrt-unprod 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

   3. sqr-neg 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

   4. sqrt-unprod 41.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

   5. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

   6. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

   7. sqrt-unprod 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

   8. add-sqr-sqrt 41.3%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   9. sqrt-unprod 70.5%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   10. sqr-neg 70.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   11. sqrt-unprod 29.2%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   12. add-sqr-sqrt 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

   13. pow1 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

   14. exp-neg 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

   15. inv-pow 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

   16. pow-prod-up 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

   17. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

   18. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

   19. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]
7. Step-by-step derivation

   1. *-rgt-identity 98.3%

      \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   2. frac-2neg 98.3%

      \[\leadsto \color{blue}{\frac{-\ell}{-e^{w}}} \]

   3. div-inv 98.3%

      \[\leadsto \color{blue}{\left(-\ell\right) \cdot \frac{1}{-e^{w}}} \]

8. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(-\ell\right) \cdot \frac{1}{-e^{w}}} \]

9. Taylor expanded in w around 0 75.7%

   \[\leadsto \left(-\ell\right) \cdot \color{blue}{\left(w \cdot \left(1 + -0.5 \cdot w\right) - 1\right)} \]

10. Final simplification 75.7%

    \[\leadsto \ell \cdot \left(1 + w \cdot \left(-1 - w \cdot -0.5\right)\right) \]
11. Add Preprocessing

Alternative 6: 70.3% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right) - \ell\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (+ l (* w (- (* w (* l 0.5)) l))))

C

double code(double w, double l) {
	return l + (w * ((w * (l * 0.5)) - l));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (w * ((w * (l * 0.5d0)) - l))
end function

Java

public static double code(double w, double l) {
	return l + (w * ((w * (l * 0.5)) - l));
}

Python

def code(w, l):
	return l + (w * ((w * (l * 0.5)) - l))

Julia

function code(w, l)
	return Float64(l + Float64(w * Float64(Float64(w * Float64(l * 0.5)) - l)))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (w * ((w * (l * 0.5)) - l));
end

Wolfram

code[w_, l_] := N[(l + N[(w * N[(N[(w * N[(l * 0.5), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.3%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. remove-double-neg 99.3%

      \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   3. associate-*l/ 99.3%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

   4. *-lft-identity 99.3%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

   5. remove-double-neg 99.3%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 43.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

   2. sqrt-unprod 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

   3. sqr-neg 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

   4. sqrt-unprod 41.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

   5. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

   6. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

   7. sqrt-unprod 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

   8. add-sqr-sqrt 41.3%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   9. sqrt-unprod 70.5%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   10. sqr-neg 70.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   11. sqrt-unprod 29.2%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   12. add-sqr-sqrt 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

   13. pow1 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

   14. exp-neg 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

   15. inv-pow 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

   16. pow-prod-up 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

   17. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

   18. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

   19. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]

7. Taylor expanded in w around 0 76.8%

   \[\leadsto \color{blue}{\ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-out 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\ell \cdot \left(-0.5 + 0.16666666666666666\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   2. add-sqr-sqrt 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   3. sqrt-unprod 58.6%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\sqrt{\ell \cdot \ell}} \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   4. sqr-neg 58.6%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \sqrt{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   5. sqrt-unprod 0.0%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\left(\sqrt{-\ell} \cdot \sqrt{-\ell}\right)} \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   6. add-sqr-sqrt 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\left(-\ell\right)} \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   7. cancel-sign-sub-inv 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   8. mul-1-neg 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\color{blue}{\left(-\left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   9. distribute-rgt-out 76.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(-\color{blue}{\ell \cdot \left(-1 + 0.5\right)}\right) - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   10. metadata-eval 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(-\ell \cdot \color{blue}{-0.5}\right) - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   11. distribute-rgt-neg-in 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\color{blue}{\ell \cdot \left(--0.5\right)} - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   12. metadata-eval 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot \color{blue}{0.5} - \ell \cdot \left(-0.5 + 0.16666666666666666\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   13. metadata-eval 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot 0.5 - \ell \cdot \color{blue}{-0.3333333333333333}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

9. Applied egg-rr 76.8%

   \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell \cdot 0.5 - \ell \cdot -0.3333333333333333\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]
10. Step-by-step derivation

    1. distribute-lft-out-- 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell \cdot \left(0.5 - -0.3333333333333333\right)\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

    2. metadata-eval 76.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot \color{blue}{0.8333333333333334}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

11. Simplified 76.8%

    \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell \cdot 0.8333333333333334\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

12. Taylor expanded in w around 0 71.6%

    \[\leadsto \ell + w \cdot \left(\color{blue}{-1 \cdot \left(w \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} - \ell\right) \]
13. Step-by-step derivation

    1. mul-1-neg 71.6%

       \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-w \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} - \ell\right) \]

    2. distribute-rgt-neg-in 71.6%

       \[\leadsto \ell + w \cdot \left(\color{blue}{w \cdot \left(-\left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} - \ell\right) \]

    3. distribute-rgt-out 71.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-\color{blue}{\ell \cdot \left(-1 + 0.5\right)}\right) - \ell\right) \]

    4. metadata-eval 71.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-\ell \cdot \color{blue}{-0.5}\right) - \ell\right) \]

    5. *-commutative 71.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-\color{blue}{-0.5 \cdot \ell}\right) - \ell\right) \]

    6. distribute-lft-neg-in 71.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \color{blue}{\left(\left(--0.5\right) \cdot \ell\right)} - \ell\right) \]

    7. metadata-eval 71.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(\color{blue}{0.5} \cdot \ell\right) - \ell\right) \]

14. Simplified 71.6%

    \[\leadsto \ell + w \cdot \left(\color{blue}{w \cdot \left(0.5 \cdot \ell\right)} - \ell\right) \]

15. Final simplification 71.6%

    \[\leadsto \ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right) - \ell\right) \]
16. Add Preprocessing

Alternative 7: 64.2% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.09:\\ \;\;\;\;-\ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.09) (- (* l w)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.09) {
		tmp = -(l * w);
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.09d0)) then
        tmp = -(l * w)
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.09) {
		tmp = -(l * w);
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.09:
		tmp = -(l * w)
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.09)
		tmp = Float64(-Float64(l * w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.09)
		tmp = -(l * w);
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.09], (-N[(l * w), $MachinePrecision]), l]

Derivation

1. Split input into 2 regimes

2. if w < -0.089999999999999997

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

      4. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

      5. remove-double-neg 100.0%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

      2. sqrt-unprod 53.2%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

      3. sqr-neg 53.2%

         \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

      4. sqrt-unprod 53.2%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

      5. add-sqr-sqrt 53.2%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

      6. add-sqr-sqrt 53.2%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

      7. sqrt-unprod 53.2%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

      8. add-sqr-sqrt 53.2%

         \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      9. sqrt-unprod 53.2%

         \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      10. sqr-neg 53.2%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      11. sqrt-unprod 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

      13. pow1 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

      14. exp-neg 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

      15. inv-pow 0.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

      16. pow-prod-up 100.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

      17. metadata-eval 100.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

      18. metadata-eval 100.0%

          \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

      19. metadata-eval 100.0%

          \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]

   7. Taylor expanded in w around 0 26.0%

      \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 26.0%

         \[\leadsto \ell + \color{blue}{\left(-\ell \cdot w\right)} \]

      2. distribute-lft-neg-out 26.0%

         \[\leadsto \ell + \color{blue}{\left(-\ell\right) \cdot w} \]

      3. *-commutative 26.0%

         \[\leadsto \ell + \color{blue}{w \cdot \left(-\ell\right)} \]

   9. Simplified 26.0%

      \[\leadsto \color{blue}{\ell + w \cdot \left(-\ell\right)} \]

   10. Taylor expanded in w around inf 26.0%

       \[\leadsto \color{blue}{-1 \cdot \left(\ell \cdot w\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 26.0%

          \[\leadsto \color{blue}{\left(-1 \cdot \ell\right) \cdot w} \]

       2. mul-1-neg 26.0%

          \[\leadsto \color{blue}{\left(-\ell\right)} \cdot w \]

   12. Simplified 26.0%

       \[\leadsto \color{blue}{\left(-\ell\right) \cdot w} \]

   if -0.089999999999999997 < w

   1. Initial program 99.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 78.9%

      \[\leadsto \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.09:\\ \;\;\;\;-\ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \ell - \ell \cdot w \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (- l (* l w)))

C

double code(double w, double l) {
	return l - (l * w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l - (l * w)
end function

Java

public static double code(double w, double l) {
	return l - (l * w);
}

Python

def code(w, l):
	return l - (l * w)

Julia

function code(w, l)
	return Float64(l - Float64(l * w))
end

MATLAB

function tmp = code(w, l)
	tmp = l - (l * w);
end

Wolfram

code[w_, l_] := N[(l - N[(l * w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.3%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. remove-double-neg 99.3%

      \[\leadsto \frac{1}{e^{\color{blue}{-\left(-w\right)}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   3. associate-*l/ 99.3%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{-\left(-w\right)}}} \]

   4. *-lft-identity 99.3%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{-\left(-w\right)}} \]

   5. remove-double-neg 99.3%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{\color{blue}{w}}} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 43.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}\right)}}{e^{w}} \]

   2. sqrt-unprod 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{w \cdot w}}}\right)}}{e^{w}} \]

   3. sqr-neg 84.6%

      \[\leadsto \frac{{\ell}^{\left(e^{\sqrt{\color{blue}{\left(-w\right) \cdot \left(-w\right)}}}\right)}}{e^{w}} \]

   4. sqrt-unprod 41.3%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}}\right)}}{e^{w}} \]

   5. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{-w}}\right)}}{e^{w}} \]

   6. add-sqr-sqrt 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w}} \cdot \sqrt{e^{-w}}\right)}}}{e^{w}} \]

   7. sqrt-unprod 84.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(\sqrt{e^{-w} \cdot e^{-w}}\right)}}}{e^{w}} \]

   8. add-sqr-sqrt 41.3%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{-w} \cdot \sqrt{-w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   9. sqrt-unprod 70.5%

      \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{\left(-w\right) \cdot \left(-w\right)}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   10. sqr-neg 70.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\sqrt{\color{blue}{w \cdot w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   11. sqrt-unprod 29.2%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot e^{-w}}\right)}}{e^{w}} \]

   12. add-sqr-sqrt 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{e^{\color{blue}{w}} \cdot e^{-w}}\right)}}{e^{w}} \]

   13. pow1 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{1}} \cdot e^{-w}}\right)}}{e^{w}} \]

   14. exp-neg 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{\frac{1}{e^{w}}}}\right)}}{e^{w}} \]

   15. inv-pow 54.5%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{1} \cdot \color{blue}{{\left(e^{w}\right)}^{-1}}}\right)}}{e^{w}} \]

   16. pow-prod-up 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{{\left(e^{w}\right)}^{\left(1 + -1\right)}}}\right)}}{e^{w}} \]

   17. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{{\left(e^{w}\right)}^{\color{blue}{0}}}\right)}}{e^{w}} \]

   18. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\left(\sqrt{\color{blue}{1}}\right)}}{e^{w}} \]

   19. metadata-eval 98.3%

       \[\leadsto \frac{{\ell}^{\color{blue}{1}}}{e^{w}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{e^{w}} \]

7. Taylor expanded in l around 0 98.3%

   \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]

8. Taylor expanded in w around 0 62.7%

   \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
9. Step-by-step derivation

   1. mul-1-neg 62.7%

      \[\leadsto \ell + \color{blue}{\left(-\ell \cdot w\right)} \]

   2. sub-neg 62.7%

      \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

10. Simplified 62.7%

    \[\leadsto \color{blue}{\ell - \ell \cdot w} \]
11. Add Preprocessing

Alternative 9: 57.2% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ \ell \end{array} \]

FPCore

(FPCore (w l) :precision binary64 l)

C

double code(double w, double l) {
	return l;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l
end function

Java

public static double code(double w, double l) {
	return l;
}

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

function tmp = code(w, l)
	tmp = l;
end

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.3%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Add Preprocessing

3. Taylor expanded in w around 0 56.3%

   \[\leadsto \color{blue}{\ell} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))